Remark 1.1 Ehresmann semigroups arose within the York School of semigroup theory led by J. B. Fountain [1]}.
The paper [2]} was particularly influential;
in particular, the conditions below which we refer to below as `deterministic' and `codeterministic'
were first introduced in this paper.
The theme of this School became that of determining which properties of (von Neumann)
regular semigroups could be generalized to a non-regular setting.
There was a particular emphasis on the non-regular
generalizations of inverse semigroups with the focus of attention being
on the abstract relationship between an element \(a\)  and its idempotents \(a^{-1}a\)  and \(aa^{-1}\) .
Subsequently, these ideas were then developed not only for semigroups but also for categories [3]}.
My own paper [4]} was written within the York School framework with the goal being
to describe the most general class of non-regular semigroups that could be regarded as natural generalizations of inverse semigroups.
This was achieved by developing ideas due to Charles Ehresmann and his students.
The key stumbling block became that of order:
in an inverse semigroup, the natural partial order encodes algebraically the order induced by subset-inclusion.
For more general classes of Ehresmann semigroups, this neat relationship between algebra and order does not hold.
Trying to deal with this problem led to the notions of `bideterministic element' and `partial isometry'
which play major rôles in this paper.

